Energy of a fluid in motion

The total energy of a fluid in motion is made up of a number of components. For unit mass of fluid and neglecting changes in magnetic and electrical energy, the magnitutes of the various forms of energy are as follows. 

This represents the work which must be done in order to introduce the fluid, without change in volume, into the system. It is therefore given by the product Pv, where F is the pressure of the system and v is the volume of unit mass of fluid. 

The potential energy of the fluid, due to its position in the earth s gravitational field, is equal to the work which must be done on it in order to raise it to that position from some 

The fluid possesses kinetic energy by virtue of its motion with reference to some arbitrarily fixed body, normally taken as the earth. If the fluid is moving with a velocity u, the kinetic energy is u /2. 

The total energy of unit mass of fluid is, therefore  

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The total energy of a fluid in motion consists of the following components internal, potential, pressure and kinetic energies. Each of these energies may be considered with reference to an arbitrary base level. It is also convenient to make calculations on unit mass of fluid. 

The governing flow equation describing flow through as porous medium is known as Darcy s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion. 

By measuring the kinetic effects of the fluid in motion, since at a given pressure drop, low-density steam will move at a much greater velocity than will high-density condensate, and the conversion of pressure energy into kinetic energy can be used to position a valve. 

For low-energy photons, the rotational and orbital motions occupy the same region in 3D space, where the same preonic fluid participates in both motions. Then, as a first approximation, it is assumed that coT = coe = co.13 Substituting in (95), the total kinetic energy of a photon in state n is... 

It is a common observation that, usually, polymer solutions flow more slowly through a tube than do pure solvents, under the same pressure. Viscosity is a measure of the energy dissipated by a fluid in motion as it resists an applied shearing force. Solution viscosity is a measure of the resistance to flow and thus, for polymer solutions, it is directly related to the size of the macromolecules and can be used to characterize the molecular weight of polymers. Practically, the flow of a polymer solution is measured and compared to the flow of the solvent alone. The ratio of the viscosity of a polymer solution, r, to that of the solvent, r , or relative viscosity, ri is used to define an important parameter of a polymer in a given solvent, the intrinsic viscosity [ii] (Eq. 40). 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

It is necessary to be able to calculate the energy and momentum of a fluid at various positions in a flow system. It will be seen that energy occurs in a number of forms and that some of these are influenced by the motion of the fluid. In the first part of this chapter the thermodynamic properties of fluids will be discussed. It will then be seen how the thermodynamic relations are modified if the fluid is in motion. In later chapters, the effects of frictional forces will be considered, and the principal methods of measuring flow will be described. 

In a stationary fluid the pressure is exerted equally in all directions and is referred to as the static pressure. In a moving fluid, the static pressure is exerted on any plane parallel to the direction of motion. The pressure exerted on a plane at right angles to the direction of flow is greater than the static pressure because the surface has, in addition, to exert, sufficient force to bring the fluid to rest. This additional pressure is proportional to the kinetic energy of the fluid it cannot be measured independently of the static pressure. 

Viscosity is the property of a fluid that offers resistance to the relative motion of fluid molecules. The energy loss due to friction in a flowing fluid is due to its viscosity. 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

Convection involves the transfer of heat by means of a fluid, including gases and liquids. Typically, convection describes heat transfer from a solid surface to an adjacent fluid, but it can also describe the bulk movement of fluid and the associate transport of heat energy, as in the case of a hot, rising gas. Recall that there are two general types of convection forced convection and natural (free) convection. In the former, fluid is forced past an object by mechanical means, such as a pump or a fan, whereas the latter describes the free motion of fluid elements due primarily to density differences. It is common for both types of convection to occur simultaneously in what is termed mixed convection. In such instance, a modified form of Fourier s Law is applied, called Newton s Law of Cooling, where the thermal conductivity is replaced with what is called the heat transfer coefficient, h ... 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

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